论文标题
调查晶体和加泰罗尼亚函数的Schur积极性
Demazure crystals and the Schur positivity of Catalan functions
论文作者
论文摘要
加泰罗尼亚函数是标志品种某些矢量束的分级欧拉（Euler）特性，是一类丰富的对称函数，其中包括$ k $ -schur函数和抛物线Hall-Littlewood多项式。我们证明，通过分区权重索引的加泰罗尼亚功能是$ u_q（\ wideHat {\ mathfrak {\ mathfrak {sl}} _ \ ell）$ - 普遍化的蛋白酶晶体，如Lakshmibai-Littelmann-Magyar和Naoi所研究的。我们获得了这些功能的Schur阳性公式，定居于Chen-Haiman和Shimozono-Weyman的猜想。我们的方法通常通过将它们与普遍的氮杂晶体的特征匹配，从而为某些矢量束的分级欧拉特征提供了关键的正式。
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_q(\widehat{\mathfrak{sl}}_\ell)$-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.
